Fredholm integration

Secondly, the linearized inverse problem is, as well as known, ill-posed because it involves the solution of a Fredholm integral equation of the first kind. The solution must be regularized to yield a stable and physically plausible solution. In this apphcation, the classical smoothness constraint on the solution [8], does not allow to recover the discontinuities of the original object function. In our case, we have considered notches at the smface of the half-space conductive media. So, notche shapes involve abrupt contours. This strong local correlation between pixels in each layer of the half conductive media suggests to represent the contrast function (the object function) by a piecewise continuous function. According to previous works that we have aheady presented [14], we 2584... 

In this subsec tion is considered a method of solving numerically the Fredholm integral equation of the second land ... 

Another school has also developed and attempted to understand the functional dependence of adsorption on heterogeneous surfaces on the vapor pressure and temperature. Various empirical or semiempirical equations were proposed [24-26] and used later to represent experimental data and to evaluate EADF by inverting Eq. (1), which belongs to the class of linear Fredholm integrals of the first kind [27]. 

Effectively, this constitutes a Fredholm integral equation of the first kind for exp[—f3G r) where we know the left-hand side, exp(—/M.4(7,)) =... 

The solution must satisfy a homogeneous Fredholm integral of the second kind with Hermitian kernel ... 

J. G. McWhirter and E. R. Pike, On the numerical inversion of the Laplace transform and similar Fredholm integral equations of the first kind, J. Phys. A Math. Gen. 11, 1729-1745 (1978). 

Let us again consider the convolution integral. Equation (86) is an example of a Fredholm integral equation of the first kind. In such equations the kernel can be expressed as a more-general function of both x and x ... 

It is now possible to see that the matrix formulation has the potential for describing the more-general Fredholm integral equation. This equation corresponds in spectroscopy to the situation where the functional form of s(x) varies across the spectral range of interest. In these circumstances, s is expressed as a function of two independent variables. Although we proceed with the present treatment formulated in terms of convolutions, the reader should bear the generalization in mind. 

Past Methods Used To Solve The Fredholm Integral Equation... 

To calculate the free energy distributions (/(AG)) of ion adsorption, the Langmuir equation was used as the kernel of the Fredholm integral equation of the first kind... 

The factor of one half appears because of a property of the Dirac delta function which is used in the derivation of Eq. (105). See also Duplantier [35] for another interpretation). Thus, if the surface charge is specified on the boundary then Eq. (Ill) is a Fredholm integral equation of the second kind [90] for the unknown potential at boundary points s. On the other hand, if the boundary potential is known then either Eq. (Ill) is used as a Fredholm integral equation of the first kind for the surfaces charge, n Vt/z, or the gradient of Eq. (105) evaluated on the boundary gives rise to a Fredholm equation... 

Equation (9.32) is a linear Fredholm integral equation of the first kind. It is also known as an unfolding or deconvolution equation. One can preanalyze the data and try to solve this first-kind integral equation. Besides the complexity of this equation, there is a paucity of numerical methods for determining the unknown function / (h) [208,379] with special emphasis on methods based on the principle of maximum entropy [207,380]. The so-obtained density function may be approximated by several models, gamma, Weibull, Erlang, etc., or by phase-type distributions. 

Atkinson, K. E. A Survey of Numerical Methods for the Solution of Fredholm Integral Equations of the Second Kind, SIAM, Philadelphia (1976). 

Let us finally briefly consider the Inverse problem how to find /(AadgU) if Vlp.T] is known from experiment, and the local isotherm is known (or assumed). Mathematically this procedure is equivalent to solving so-called Fredholm integrals of the first kind the local isotherm functions serve as the kernel. Computationally speaking, this is an ill-posed problem because minor variations in the data may cause substantial variations in /(Aads )- n other words, the Inevitable experimental noise thwarts one obtaining a physically significant distribution. In practice the elaboration may follow one of three paths ... 

This is a Fredholm integral equation of the first kind. The regularized solution to this equation has been applied to the measurement both for the moments and the size distribution of a wide range of latices [46]. K has been given by van de Hulst [45] in terms of particle size/refractive index domain. Mie theory applies to the whole domain but in the boundary regions simpler equations have been derived. 

For r e r, this relationship leads to a Fredholm integral equation of the second type for the field Ey within the region F. The Green s function for the background sequence, Gb, which is present in equation (9.21) can be calculated from a simple recursive formula. Details about this procedure can be found in a monograph by Berdichevsky and Zhdanov (1984). 

At this level of approximation, the problem is to invert the Fredholm integral of eq. (1) or (2). This is an ill-posed problem in general. The usual method of solution is to assign a functional form to the distribution function (//) otJ[s), such as a multimodal gamma distribution, and then fit the parameters in this function to a least squares match to the experimental isotherm. In addition to limiting the treatment to only one kind of heterogeneity, eqs. (1) and (2) omit any effects of networking or pore connectivity. 

Equation 1 represents a Fredholm integral and its inversion is well known to present an ill-posed problem. Since we are only interested in the numerical values of f(H), we can rewrite equation 1 as a summation ... 

The goal, then, is to determine the distribution function m0(r, t) for each voxel so that the intrinsic magnetization may be determined using Eq. (12). In what follows, the explicit dependence on position is dropped, with the understanding that the same analysis applies to each voxel throughout the sample. Equation (11) can be rewritten as a Fredholm integral equation of the first kind ... 

From the mathematical point of view, equation (1) is a linear Fredholm integral equation of the first kind, which can be written in a more general form as follows ... 

Whatever the chosen calculation process, the isotherm analysis is based on a simple physical model describing, in the simplest way, the global experimental isotherm as a sum of partial isotherms corresponding to homogeneous adsorption patches. Hence, the amount of adsorbed molecules (probe) is given by the following Fredholm integral equation ... 

At the level of approximation invoked by the simple geometric model, the mathematical problem becomes one of inverting Eq. (14), a linear Fredholm integral equation of the first kind, to obtain the PSD. The kernel r(P, e) represents the thermodynamic adsorption model, r(P) is the experimental function, and the pore size distribution /(//) is the unknown function. The usual method of determining /(H) is to solve Eq. (14) numerically via discretization into a system of linear equations. 

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